Introduction to Martingale Problems

Abstract : The generator for a Markov process is a linear operator that characterizes infinitesimally the evolution of the distribution of the process. Classically, the Hille-Yosida theory of operator semigroups was used to connect the Markov process to its generator. In the context of diffusion processes, Stroock and Varadhan showed that Markov processes can be characterized by the requirement that certain functionals of the process, determined by the generator, must be martingales, that is, the Markov process can be characterized as the unique solution of a martingale problem. For example, Brownian motion is the unique solution of the martingale problem for the Laplacian.

Martingale problems provide a powerful approach to the study of Markov processes. The basic theory of martingale problems will be reviewed and several illustrative examples of its application will be given.

NOTE: This talk is intended to give some useful background for the November 12 talk in the Complex Systems Seminar.